Tuple domination on graphs with the consecutive-zeros property (original) (raw)

The 2-Tuple Domination Problem on Circular-arc Graphs

Journal of Mathematics and Informatics

Given a simple graph) , (= E V G and a fixed positive integer k. In a graph G , a vertex is said to dominate itself and all of its neighbors. A set V D ⊆ is called a k-tuple dominating set if every vertex in V is dominated by at least k vertices of D. The k-tuple domination problem is to find a minimum cardinality k-tuple dominating set. This problem is NP-complete for general graphs. In this paper, the same problem restricted to a class of graphs called circular-arc graphs are considered. In particular, we presented an) (2 n O-time algorithm to solve the 2-tuple domination problem on circular-arc graphs, one of the non-tree type graph classes.

Efficient algorithms for tuple domination on co-biconvex graphs and web graphs

arXiv: Combinatorics, 2020

A vertex in a graph dominates itself and each of its adjacent vertices. The kkk-tuple domination problem, for a fixed positive integer kkk, is to find a minimum sized vertex subset in a given graph such that every vertex is dominated by at least k vertices of this set. From the computational point of view, this problem is NP-hard. For a general circular-arc graph and k=1k=1k=1, efficient algorithms are known to solve it (Hsu et al., 1991 & Chang, 1998) but its complexity remains open for kgeq2k\geq 2kgeq2. A 0,10,10,1-matrix has the consecutive 0's (circular 1's) property for columns if there is a permutation of its rows that places the 0's (1's) consecutively (circularly) in every column. Co-biconvex (concave-round) graphs are exactly those graphs whose augmented adjacency matrix has the consecutive 0's (circular 1's) property for columns. Due to A. Tucker (1971), concave-round graphs are circular-arc. In this work, we develop a study of the kkk-tuple domination problem on ...

On the complexity of {k}-domination and k-tuple domination in graphs

Information Processing Letters, 2015

We consider two types of graph domination-{k}-domination and k-tuple domination, for a fixed positive integer k-and provide new NP-complete as well as polynomial time solvable instances for their related decision problems. Regarding NP-completeness results, we solve the complexity of the {k}-domination problem on split graphs, chordal bipartite graphs and planar graphs, left open in 2008. On the other hand, by exploiting Courcelle's results on Monadic Second Order Logic, we obtain that both problems are polynomial time solvable for graphs with clique-width bounded by a constant. In addition, we give an alternative proof for the linearity of these problems on strongly chordal graphs.

Efficient and Perfect domination on circular-arc graphs

Electronic Notes in Discrete Mathematics, 2015

Given a graph G = (V, E), a perfect dominating set is a subset of vertices V ′ ⊆ V (G) such that each vertex v ∈ V (G) \ V ′ is dominated by exactly one vertex v ′ ∈ V ′. An efficient dominating set is a perfect dominating set V ′ where V ′ is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remains NP-Hard, even when restricted to certain graphs families. We study both variants of the problems for the circular-arc graphs, and show efficient algorithms for all of them.

On the algorithmic complexity of -tuple total domination

Discrete Applied Mathematics, 2014

For a fixed positive integer k, a k-tuple total dominating set of a graph G is a set D ⊆ V (G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies ktuple total domination from an algorithmic point of view. In particular, we present a lineartime algorithm for the k-tuple total domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph, which include trees, block graphs, cacti and block-cactus graphs. We also establish NP-hardness of the k-tuple total domination problem in undirected path graphs.

A simple linear algorithm for the connected domination problem in circular-arc graphs

Discussiones Mathematicae Graph Theory, 2004

A connected dominating set of a graph G = (V, E) is a subset of vertices CD ⊆ V such that every vertex not in CD is adjacent to at least one vertex in CD, and the subgraph induced by CD is connected. We show that, given an arc family F with endpoints sorted, a minimum-cardinality connected dominating set of the circular-arc graph constructed from F can be computed in O(|F |) time.

Upper k-tuple domination in graphs

Discrete Mathematics & Theoretical Computer Science

Graph Theory For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V (G) such that |N [v] ∩ S| ≥ k for every vertex v, where N [v] = {v} ∪ {u ∈ V (G) : uv ∈ E(G)}. The upper k-tuple domination number of G, denoted by Γ×k (G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ×k (G) for r-regular graphs G with r ≥ k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ×2 (G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs.

Roman k-Tuple Domination in Graphs

Iranian Journal of Mathematical Sciences and Informatics, 2020

For any integer k ≥ 1 and any graph G = (V,E) with minimum degree at least k−1, we define a function f : V → {0, 1, 2} as a Roman k-tuple dominating function on G if for any vertex v with f(v) = 0 there exist at least k and for any vertex v with f(v) 6= 0 at least k − 1 vertices in its neighborhood with f(w) = 2. The minimum weight of a Roman k-tuple dominating function f on G is called the Roman k-tuple domination number of the graph where the weight of f is f(V ) = ∑ v∈V f(v). In this paper, we initiate to study the Roman k-tuple domination number of a graph, by giving some sharp bounds for the Roman k-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman k-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al. [1] in 2004 for the Roman domination number.

k-tuple total domination in graphs

2010

A set S of vertices in a graph G is a k-tuple total dominating set, abbreviated kTDS, of G if every vertex of G is adjacent to least k vertices in S. The minimum cardinality of a kTDS of G is the k-tuple total domination number of G. For a graph to have a kTDS, its minimum degree is at least k. When k = 1, a k-tuple total domination number is the well-studied total domination number. When k = 2, a kTDS is called a double total dominating set and the k-tuple total domination number is called the double total domination number. We present properties of minimal kTDS and show that the problem of finding kTDSs in graphs can be translated to the problem of finding k-transversals in hypergraphs. We investigate the k-tuple total domination number for complete multipartite graphs. Upper bounds on the k-tuple total domination number of general graphs are presented.

K-Tuple and K-Tuple Total Dominations on Web Graphs

Matemática Contemporânea, 2022

In this work we address k-tuple and k-tuple total dominations on the subclass of circular-arc graphs given by web graphs. For the non total version, we present a linear time algorithm based on the regularity of the closed neighborhoods associated with web graphs which allows the use of modular arithmetic for integer numbers. For the total version, we derive bounds for this graph class. 1 Preliminaries Domination in graphs is useful in different applications. There exist many variations-such as k-tuple domination and k-tuple total domination, among others-regarding slight differences in their definitions. These differences make circular-arc graph subclasses adequate and useful mostly due to their relation to "circular" issues, such as in forming sets of representatives, in resource allocation in distributed computing systems,